In this paper we deal with complete linear Weingarten hypersurfaces immersed into Riemannian space forms. Assuming an Okumura type inequality on the total umbilicity tensor of such hypersurfaces, we prove that either the hypersurface is totally umbilical or it holds an estimate for the norm of the total umbilicity tensor, which is also shown be sharp in the sense that the product of space forms realize them. Our approach is based on a version of the Omori–Yau maximum principle for a suitable Cheng–Yau type operator.
Our purpose in this paper is to study the stability of f-maximal spacelike hypersurfaces immersed in a weighted generalized Robertson-Walker spacetime −I ×ρM n f , where M n f is a weighted Riemannian manifold endowed with a weight function f. In this setting, we obtain sufficient conditions to guarantee that an f-maximal hypersurface be L f-stable, where L f stands for the weighted Jacobi operator. 1. Introduction. Let (M n+1 , ,) be an orientable (n+1)-dimensional Lorentzian manifold endowed with a timelike vector field V and let f : M n+1 → R be a smooth function. The weighted Lorentzian manifold M n+1 f associated with M n+1 and f is the triple (M n+1 , , , e −f dM), where dM denotes the standard volume element of M n+1 induced by the metric ,. We will refer to the function f as being the weight function associated to M n+1 f. In this setting, an important tensor is the Bakry-Émery Ricci tensor Ric f , a natural generalization of the Ricci tensor Ric of M n+1 f defined by (1.1) Ric f = Ric + Hess f, where Hess f is the Hessian of f on M n+1 f. Appearing naturally in the study of self-shrinkers, Ricci solitons, harmonic heat flows and many other subjects in differential geometry, weighted
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